Reliable positioning services are extremely important for users in mountainous environments. However, in such environments, the service reliability of conventional wireless positioning technologies is often disappointing due to frequent nonline-of-sight (NLoS) propagation and poor geometry of available anchor nodes. Hence, we propose a unmanned aerial vehicle (UAV)-enabled positioning system that utilizes UAV's mobility to overcome the above challenges. In this article, we first analyze and model the major causes of service failures in the proposed system. In particular, a geometry-based NLoS probability model is established based on the digital elevation models (DEM) of realistic terrain for reliability analysis. Subsequently, we propose a reliability prediction method and derive the corresponding metric to evaluate the system's ability to provide reliable positioning services. Moreover, we also develop a voting-based method for the further enhancement of service reliability. Monte-Carlo simulations show that in mountainous environments, the proposed reliability prediction method could achieve a prediction accuracy that is at least 36.8% higher than that of the existing technique. In addition, in the experiments conducted in two typical valley scenarios, the proposed reliability enhancement method improves the service reliability of the proposed system by 23% and 29%, respectively. These numerical results demonstrate the strong potential of the proposed system and methods for reliable positioning.